Is there a way to set the gzip compression at the web.config level or can I only do this in the IIS management console?
Setting changes are required within IIS. This is an old but good article that should help: http://totaldevpro.com/blogs/dotnetarticles/pages/enabling-http-compression-in-iis-6-0.aspx
Yes you can enable compression with the web.config, as the article below shows- but it can depend on the permissions on the server allows sites.
You should note that dynamic compression (anything that needs to be processed before ti can be sent to the client) can increase the load on the server because its having to do compression on every single request.
Edit: note this is for IIS7 (as you have tagged)
I managed to enable GZip compression using web.config file on Windows Shared Hosting at GoDaddy. Check out my blog post below:
How to Gzip on ASP.NET and GoDaddy
Thanks.